We probed pieces of flat-mounted tiger salamander retina with a stimulus composed of twenty small spots flashed one-at-a-time for 50 ms () at locations within a 200–300 μm diameter area, roughly equal to the size of a typical salamander ganglion cell’s receptive field center (Segev et al., 2004
). The responses were measured by whole-cell voltage clamp of pairs of ganglion cells located in the same region held near the Cl−
reversal potential (). Spots were flashed 60 times each in random order on a rod-suppressing background. Their location, color, size and intensity were chosen to preferentially stimulate 1–2 long-wavelength sensitive (L) cones at each location (see Materials and Methods
We ask, given the responses of the ganglion cells, how well can one distinguish among the different spots, how does this performance depend upon the number of ganglion cells used, and what is the contribution of local irregularities, as compared to the contribution of variations in receptive field position, size and orientation?
One possibility is that because the spots were all, nominally, within the center of the receptive fields of both ganglion cells, the responses of the two ganglion cells to all of the spots should be highly correlated. This would make the information encoded by the ganglion cells highly redundant. The responses of two ganglion cells are shown in . The flashes elicited a brief, rapidly increasing inward current, followed by a slower decay. The responses to different spots were not highly correlated between the two ganglion cells: spot B elicited large responses from both cells, while spots A and C elicited small responses from ganglion cell 1 and large responses from ganglion cell 2. Under these conditions, spots typically evoked short bursts of spikes in a current-clamped ganglion cell (), and the magnitude of the input current response to a spot presented in a given location was roughly correlated with the number of spikes evoked by that spot ().
The input current responses of a given ganglion cell were highly stereotyped, with most of the variation among responses accounted for by uniform scaling of the entire response waveform. We used principal components analysis to quantify this response (). Among all ganglion cells, the first principal component resembled the mean response and, on average, accounted for 94.3±1.3% (mean±s.e.m.) of the signal variance. The second largest principal component captured, on average, 2.3±0.6% of the signal variance, the third 1.0±0.3%, with the remainder distributed amongst the remaining components. Because a large fraction of the variation was in the first principal component with very small contributions from all of the remaining components combined, the amplitude of a given response was defined as the projection of a given response waveform along the first principal component for that ganglion cell (see Materials and Methods).
If the responses of the ganglion cells were completely redundant, the joint responses of the two cells should be correlated and fall along a straight line in a plot of the second cell’s response amplitude versus the first cell’s. However, many of the responses of this pair of cells deviated significantly from this pattern (). Considerable decorrelation was observed in most of the recorded ganglion cell pairs (). This suggests that despite the closeness of their cell bodies, the receptive fields of neighboring ganglion cells do not completely overlap. Whether this is caused by a gross difference in receptive field position and size or local irregularities within otherwise overlapping receptive fields is considered below.
Decorrelation should make the responses of neighboring ganglion cells more independent and allow more information to be represented by many cells. The independence of visual encoding can be calculated using the mutual information between stimulus and response, i.e. how much information is conveyed by the responses about which of the twenty spots was flashed. If two ganglion cells convey information independently, the information about the stimulus calculated from the joint response should be equal to the sum of the mutual information of each ganglion cell taken alone. For this pair of cells, the joint mutual information (1.45±0.05 bits, mean±s.e.m.) was slightly higher than expected if the cells signaled independently (1.37±0.05 bits, ), indicating that the pair signals slightly synergistically. The pair conveys a substantial fraction (34%) of the maximum possible information about spot location (log2 20 ~4.3 bits).
The signaling among most ganglion cell pairs was nearly independent, i.e.
the measured mutual information for ganglion cells pairs fell close to that predicted by summing the information of each ganglion cell taken alone (). Simultaneously recorded pairs (black points) and pairs recorded at different times but from the same patch of retina (gray points) were distributed similarly, suggesting that the contribution of correlated trial-by-trial variation (noise correlation) was small. The majority of pairs fell slightly below the unity line (, coarse dashed line) with slope 0.95 (solid lines), indicating that the signaling among pairs was mildly redundant, close to the 10% redundancy reported previously for spikes trains from pairs of ganglion cells under naturalistic stimulation (Puchalla et al., 2005
Figure 4 Most ganglion cell pairs signal nearly independently. The mutual information for a pair of ganglion cells is plotted as function of the predicted mutual information assuming independence. The coarse dashed line has slope one, the fine dashed line has (more ...)
One possible explanation for the near-independence of ganglion cell signaling is that the relatively low redundancy was due to noise, which, if large enough, would have caused the responses of ganglion cells to be independent. To assess the contribution of noise, we calculated the mutual information for a synthetic pair composed of a single cell sampled twice – the equivalent of two cells with completely overlapping receptive fields but independent noise (, blue points). If there were no noise in the system, these points would lie along the line of slope one-half (fine dashed line). As expected, noise causes the responses to be more independent, but does not account for the near-independence of most ganglion cell pairs. Of the 105 pairs shown, only one ganglion cell pair was as redundant as expected from completely overlapping receptive fields (, cyan arrowhead). The near independence of ganglion cell pairs is therefore the result of genuine differences in spatial sampling, not noise in retinal circuitry.
We divided our ganglion cell population into three functional types based on their reverse correlation to flickering checkerboard stimulation – those in which an increase in light intensity caused an inward, outward or mixed current response (). There was no significant difference in the information encoded among the three functional classes (, e.g. inward vs. outward, p=0.14 by Kolmogorov-Smirnov test). There was no significant difference in the mean independence among pairs composed of different types (), with the exception of outward-outward pairs, which were more redundant, signaling with 90% independence compared to the 94–99% independence of the other pairs. These results indicate that under our stimulus and recording conditions, information about the location of small flashed spots was distributed widely within the ganglion cell population and represented in a similar fashion by different cell types. Because of this similarity, we pooled results across cell types in the remainder of the paper. It is certainly possible that under different stimulus conditions, light responses may differ more markedly among cell types.
Figure 5 Analysis of receptive field spatial autocorrelation and information among cell types. a, Examples of inward, mixed and outward-type receptive fields. For clarity, each trace represents the average linear filter waveform of the 2×2 underlying linear (more ...)
To characterize the local irregularities in our receptive fields, the spatial autocorrelation function of the receptive field was calculated, taking into account the autocorrelation due to the size of the checkerboard squares (20 μm, for details see Materials and Methods
). The width of this function summarizes the typical size of spatial variations in the sensitivity of the receptive field. The spatial autocorrelation function was well described by the sum of two Gaussian profiles with half-widths of 21 and 83 μm (). There was no significant difference in the fit among the three functional cell types (data not shown). The smaller width is comparable to the spacing between cone photoreceptors (~20 μm), and the larger is consistent with the range of receptive field center diameters (~100–400 μm) reported in salamander (Segev et al., 2004
In retinal mosaics, variability in the position and size of receptive fields can have a substantial effect on the representation of spatial detail (Liu et al., 2009
). To measure this contribution in groups of ganglion cells with highly overlapping receptive fields, the measured receptive field of each ganglion cell was fitted with a two-dimensional Gaussian (; Materials and Methods
). The fitted receptive fields were on average 203±107 μm in diameter with average long/short axis ratio of 1.5, consistent with receptive fields generated from spike-triggered averages (Segev et al., 2004
). This diameter roughly matched the wider component of the spatial autocorrelation function (). Local irregularities can be seen as peaks and valleys in the sensitivity profile of individual ganglion cells, which are not captured by a Gaussian fit ().
Figure 6 Receptive field measurement and fitting with Gaussian receptive field model. a, Two-dimensional receptive field map for two ganglion cells (red and blue) with sensitivity represented on an intensity scale. Ellipses are the one-standard-deviation boundaries (more ...)
Next, we tested whether the receptive field model could account for the actual coding performance of ganglion cells in the flashed-spot experiment. Because the mutual information depends both on signal and noise, we needed to have a model of the noise in the response, i.e. how the variability in a response to a given spot depended on the mean amplitude of the response to that spot. For nearly all ganglion cells, the measured noise increased approximately linearly with mean signal amplitude with a slope in the range of 0.1 to 0.3 atop a relatively small constant background noise (). Thus, we assumed that the average response to a small flash was proportional to the Gaussian receptive field amplitude at the location of the spot and that the noise was linearly related to this signal. For each cell, the slope and offset of the fitted line was used to calculate the expected standard deviation of the noise in the response for a given response amplitude. The slope of the noise model was not significantly correlated among cells, whether recorded at the same time or in the same location (; Pearson correlation, r = 0.039).
Figure 7 Noise in retinal ganglion cells increases approximately linearly with a different characteristic slope for each ganglion cell. a, Each panel represents one ganglion cell. Each point represents the mean and standard deviation of the responses to one of (more ...)
Despite taking into account the positions and sizes of the receptive fields, and controlling for differences in noise among individual cells, the Gaussian receptive field model could not account for a substantial fraction of the information represented by the ganglion cells. For groups composed of five cells, the observed responses conveyed 33% more information than the matched, Gaussian receptive field model (2.4±0.3 bits compared to 1.8±0.5 bits, ). Furthermore, the actual performance was more consistent: although the Gaussian model predicted that many groups would encode less than 12% of the information about spot location, the actual performance fell within a smaller range, with groups never encoding less than 34% and at best encoding 81% of the information.
As expected, as more ganglion cells were considered, the information about the stimulus increased (). The increase was more rapid, however, for the measured data than the Gaussian model. Groups of eight ganglion cells on average encoded 68.0±0.4% of the information; in comparison, model cells encoded only 54.5±0.7% of the information. The increases in all cases were explained by a simple model in which the observed pairwise redundancy is uniformly distributed among all of the cells in the group (Puchalla et al., 2005
; , solid lines, see Materials and Methods
). This suggests that pairwise interactions are the dominant source of redundancy, as found in more complex stimulus conditions (Schneidman et al., 2006
; Shlens et al., 2009
Pairwise decorrelation is potentially a strong effect because the number of pairwise interactions increases quadratically with the number of cells in a group, and a modest decrease in pairwise redundancy can cause a large increase in the information conveyed by the entire group. Consistent with this prediction, the actual spatial information encoded was significantly higher for the responses of real ganglion cells than that predicted by the classical model for a groups as small as n=2 (p=0.0016).
In models of retinal mosaics, most of the increase in mutual information between stimulus and response occurred when the size and orientation of receptive fields were optimized to compensate for irregularities in the lattice itself (Liu et al., 2009
). Further optimization of the receptive field boundaries by adding non-Gaussian distortions had a relatively weak effect. In contrast, in our experiments with groups of ganglion cells with highly overlapping receptive fields, local irregularities caused an improvement nearly as large or larger than the improvement caused by de-overlapping the receptive fields and allowing them to have differing size and shape. That is, the difference between the actual performance (, black line) and the Gaussian model (, red line) is nearly as great or greater than the difference between the Gaussian model and the case of n perfectly overlapping cells (, blue line).
One possible explanation for the poor performance of the Gaussian receptive field model is that it is somehow handicapped by bias in the receptive field fitting procedure, reducing its coding capacity. In this comparison, however, the entropy of a single model cell’s response was constrained to be the same as in the measured data (see Materials and Methods), so that the coding capacity of a single model cell was equal with respect to the stimulus. The differences in signaling are thus due to differences in receptive field overlap, not the coding capacity of individual cells.
The actual performance was explained, however, if local irregularities in each cell’s spatial receptive field profile were taken into account. Another receptive field model was constructed by generating responses proportional to the actual sensitivity at the spot location, rather than the Gaussian approximation (for details, see Materials and Methods). The irregular receptive field model more closely predicted the actual performance (, green points).